Answer:
[tex]x = 3, x = -1[/tex]
Step-by-step explanation:
Given expression is
[tex]x^2e^x-2e^x - 3e^x =0[/tex]
and we are asked to find the solution set
1. Factor out [tex]e^x[/tex]:
==> [tex]e^x(x^2 - 2x - 3)=0[/tex]
2. If ab = 0 then either a = 0 or b = 0:
[tex]e^x = 0 $ or $ x^2 - 2x - 3 = 0[/tex]
3. For [tex]e^x = 0[/tex] there is no solution since the exponential function never reaches zero
4. For [tex]x^2 - 2x - 3 = 0[/tex], factor the expression:
[tex]x^2 - 2x - 3 = (x -3)(x+1)[/tex]
5. This means
[tex](x - 3)(x + 1) = 0\\\\(x - 3) = 0 == > x = 3\\\\(x + 1 ) = 0 == > x = -1\\\\[/tex]
Therefore the solution set is
[tex]x = 3, x = -1[/tex]